Question: You have found the following ages (in years) of 4 lions. The lions are randomly selected from the 31 lions at your local zoo: $ 3,\enspace 7,\enspace 5,\enspace 20$ Based on your sample, what is the average age of the lions? What is the variance? You may round your answers to the nearest tenth.
Answer: Because we only have data for a small sample of the 31 lions, we are only able to estimate the population mean and variance by finding the sample mean $({\overline{x}})$ and sample variance $({s^2})$ To find the sample mean , add up the values of all $4$ samples and divide by $4$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4$ To compensate for this underestimation, rather than simply averaging the squared deviations from the mean , we total them and divide by $n - 1$ $ {s^2} = \dfrac{\sum\limits_{i=1}^{{n}} (x_i - {\overline{x}})^2}{{n - 1}} $ $ {s^2} = \dfrac{{33.64} + {3.24} + {14.44} + {125.44}} {{4 - 1}} $ $ {s^2} = \dfrac{{176.76}}{{3}} = {58.92\text{ years}^2} $ We can estimate that the average lion at the zoo is 8.8 years old. There is a variance of 58.92 years $^2$.